Some results on fixed points of multifunctions on abstract metric spaces

Recently, Kunze et al. [H.E. Kunze, D. La Torre, E.R. Vrscay, Contraction multifunctions, fixed point inclusions and iterated multifunction system, J. Math. Anal. Appl. 330 (2007) 159-173] proved some fixed point results for multifunctions in metric spaces. Rezapour and Haghi [Sh. Rezapour, R.H. Haghi, Fixed point of multifunctions on cone metric spaces, Numer. Funct. Anal. Optim. 30 (2009) 825-832] adapted these results to the case of abstract (cone) metric spaces when the underlying cone is normal with normal constant M=1. The aim of this paper is to show that these results remain valid in the case when M>1. Introducing new contraction conditions our results generalize fixed point theorems of Covitz and Nadler, Kunze et al. and Rezapour and Haghi. An example is given to distinguish our results from the known ones. In addition, the case when two mappings are considered is treated.